Let $O$ be an order in a real quadratic number field. Let $G$ be an $S$-arithmetic subgroup of $SL_2(O)$ whose intersection with $SL_2(\mathbb Z)$ is not of finite index in $SL_2(\mathbb Z)$.

Can the fundamental domain of the action of $G$ on the complex upper half plane $\mathbb H$ have finite volume?

What if we assume $G$ to be of finite index in $SL_2(O)$?

What if we remove the hypothesis that $G\cap SL_2(\mathbb Z)$ be of infinite index in $SL_2(\mathbb Z)$?